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| For a control Cauchy problem
$$\dot x= {f}(t,x,u,v) +\sum_{\alpha=1}^m g_\alpha(x) \dot u_\alpha\quad x(a)=\bar x, $$
(notice the presence of the input's derivative) on an interval $[a,b]$, we propose the notion of Limit Solution $x$ that verifies the following properties: i) $x$ is defined for ${\mathcal L}^1$ inputs $u$ and for standard, bounded measurable, controls $v$; ii) in the commutative case (i.e. when $[g_{\alpha},g_{\beta}]\equiv 0,$ for all $\alpha,\beta=1,\dots,m$), $x$ coincides with the solution constructed via multiple fields' rectification;
iii) $x$ subsumes former concepts of solution valid for the generic, noncommutative case.
In particular, when $u$ has bounded variation, we investigate the relation between limit solutions and (single-valued) graph completion solutions.
Furthermore, we prove consistency with the classical Caratheodory solution when $u$ and $x$ are absolutely continuous.
Even though some specific problems are better addressed by means of special representations of the solutions, we believe that various theoretical and practical issues call for a unified notion of trajectory. For instance, this is the case of optimal control problems, possibly with state and end-point constraints, for which no extra assumptions (like e.g. coercivity, boundedness, commutativity) are made in advance. |
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